A simplified system of equations describing the 2-D flow of fluid of uniform depth , with an imposed temperature difference , under gravity , with buoyancy , thermal diffusivity , and kinematic viscosity . The full equations are

(1) |

(2) |

(3) |

In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified
system, periodic solutions of the form

(4) |

(5) |

Lorenz included the following terms in his system of equations,

(6) | |||

(7) | |||

(8) |

and obtained the simplified equations

(9) | |||

(10) | |||

(11) |

where

(12) | |||

(13) | |||

(14) |

Lorenz took and .

The Critical Points at (0, 0, 0) correspond to no convection, and the
Critical Points at

(15) |

(16) |

(17) |

**References**

Gleick, J. *Chaos: Making a New Science*. New York: Penguin Books, pp. 27-31, 1988.

Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.''
*Physica D* **9**, 189-208, 1983.

Lichtenberg, A. and Lieberman, M. *Regular and Stochastic Motion.* New York: Springer-Verlag, 1983.

Lorenz, E. N. ``Deterministic Nonperiodic Flow.'' *J. Atmos. Sci.* **20**, 130-141, 1963.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. *Chaos and Fractals: New Frontiers of Science.*
New York: Springer-Verlag, pp. 697-708, 1992.

Tabor, M. *Chaos and Integrability in Nonlinear Dynamics: An Introduction.* New York: Wiley, 1989.

© 1996-9

1999-05-25